We derive exact solutions to the Vakhnenko-Parkes equation by means of the complex method, and then we illustrate our main results by some computer simulations. We can apply the idea of this study to related nonlinear differential equation.

In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first, and then find all meromorphic general solutions of in combination the Newell-Whitehead equation, the NLS equation, and the Fisher equation with degree three. Our result shows that all rational and simply periodic exact solutions of the combined the Newell-Whitehead equation, NLS equation, and Fisher equation with degree three are solitary wave solutions, and the method is simpler than other methods.